Strategic bidding in a primary reserve auction

Electricity grids are subject to a constant change of demand. If a power line is overloaded, the demand is rerouted to another line, which is then also likely to overload due to the sudden spike in voltage. Due to this cascading effect a grid-wide blackout is not at all improbable; one occurred in Italy in 2003. The costs of such a blackout are immense in today’s modern society. Transport and telecommunication systems have such a high power demand that a backup power generator system would come at a very high cost. To solve this, Germany requires the electricity grid operators to have Primary Reserve Capacity on standby..

ADAPTIVE MAJORITY PROBLEMS FOR RESTRICTED QUERY GRAPHS AND FOR WEIGHTED SETS

Suppose that the vertices of a graph G are colored with two colors in an unknown way. The color that occurs on more than half of the vertices is called the majority color (if it exists), and any vertex of this color is called a majority vertex. We study the problem of finding a majority vertex (or show that none exists), if we can query edges to learn whether their endpoints have the same or different colors. Denote the least number of queries needed in the worst case by m(G). It was shown by Saks and Werman that m(K-n) = n - b(n) where b(n) is the number of 1's in the binary representation of n. In this paper we initiate the study of the problem for general graphs. The obvious bounds for a connected graph G on n vertices are n - b(n) <= m(G) <= n - 1. We show that for any tree T on an even number of vertices we have m(T) = n - 1, and that for any tree T on an odd number of vertices, we have n - 65 <= m (T) <= n - 2. Our proof uses results about the weighted version of the problem for K-n, which may be of independent interest. We also exhibit a sequence G(n) of graphs with m(G(n)) = n - b(n) such that the number of edges in G(n) is O(nb(n))

Generalized matching games for international kidney exchange

We introduce generalized matching games defined on a graph $G=(V,E)$ with an edge weighting $w$ and a partition $V=V_1 \cup \dots \cup V_n$ of~$V$. The player set is $N = \{ 1, \dots, n\}$, and player $p \in N$ owns the vertices in $V_p$. The value $v(S)$ of coalition $S \subseteq N$ is the maximum weight of a matching in the subgraph of $G$ induced by the vertices owned by players in $S$. If $|V_p|=1$ for every player~$p$ we obtain the classical matching game. We prove that checking core non-emptiness is polynomial-time solvable if $|V_p|\leq 2$ for each $p$ and co-\NP-hard if $|V_p|\leq 3$ for each $p$. We do so via pinpointing a relationship with $b$-matching games and also settle the complexity classification on testing core non-emptiness for $b$-matching games. We propose generalized matching games as a suitable model for international kidney exchange programs, where the vertices in $V$ correspond to patient-donor pairs and each $V_p$ represents one country. For this setting we prove a number of complexity results

Edge Ordered Turan Problems

We introduce the Turan problem for edge ordered graphs. We call a simple graph edge ordered, if its edges are linearly ordered. An isomorphism between edge ordered graphs must respect the edge order. A subgraph of an edge ordered graph is itself an edge ordered graph with the induced edge order. We say that an edge ordered graph G avoids another edge ordered graph H, if no subgraph of G is isomorphic to H. The Turan number ex(<)'(n, H) of a family H of edge ordered graphs is the maximum number of edges in an edge ordered graph on n vertices that avoids all elements of H.We examine this parameter in general and also for several singleton families of edge orders of certain small specific graphs, like star forests, short paths and the cycle of length four